Inequalities for probabilities of joint occurrence of several events are important in combinatorial analysis, probability theory, and many applications. This paper describes a method for constructing upper and lower bounds for probabilities of simultaneous occurrence of r out of n events. The method uses different representations of the probabilities as sums and estimates the terms separately. This yields inequalities that are more accurate than the earlier bounds and corresponding to trivial representations. The resulting new inequalities are optimal. There are examples showing that these inequalities can become equalities. Similar inequalities have been proven for conditional probabilities of corresponding events with respect to some σ-field. Averaging of both sides of inequalities for conditional probabilities can yield more accurate bounds of unconditional probabilities.